15 Puzzle

The 15 Puzzle dates from 1880 and it has played the same trick on everyone ever since: the first ten tiles slot in almost by themselves, the last two look impossible. They are not — you just have to stop improvising and follow an order.

The rules

A grid of numbered tiles and one empty square. You can only move a tile adjacent to the gap, which really means: the gap moves, and the tiles go round it. You win when the numbers are in order, left to right and top to bottom, with the gap at the end.

In the physical puzzle half of all possible arrangements have no solution. Sam Loyd built his fame on that, offering a thousand dollars for a position he knew was impossible — except the puzzle was not his, he claimed authorship years later.

On Pocket Games

There are three sizes, one per difficulty, and each starts from a grid scrambled a different amount:

The scramble starts from the solved grid and slides real tiles, never immediately undoing the previous slide: that is why every game has a solution.

• Moves are counted per push. Tap a tile on the same row or column as the gap, even a distant one, and every tile in between slides with it, as a single move.
• On a computer you play with the arrow keys: the arrow shows where the tile goes, and each press moves one tile.
• Record: the lowest move count for each size, stored on your account. It is the only thing that syncs.
• Game in progress: from the first move on, the grid is saved after every slide, and you pick it up from the start screen with the moves already made.
• No undo: you fix a slip by moving again, and that move counts.

The method: top to bottom, left to right

Everyone’s mistake is shoving tiles around until the mess moves somewhere else. The way through is different: finish one row and never touch it again.

1. The first row. Put 1, 2 and 3 in place, then deal with the 4. When you move on to the next row, pretend the first one does not exist: every manoeuvre from then on happens below it.

2. The middle rows. Same thing, always working only in the unfinished part, which gets shorter each time.

3. When two rows are left, change direction: solve them by columns, two tiles at a time, from the left. The last two rows can no longer be handled one at a time, because finishing the last row would break the one above it.

4. The final two tiles go in with one full turn of the gap around the closing 2×2 square.

The last-pair manoeuvre

This is where almost everyone gives up. Take the first row of a 4×4: the 4 belongs at the far right, but the moment you put it there, moving the 3 into place pushes it away. The answer is not to place them one at a time:

• Put the 3 where the 4 should go, at the end of the row.
• Put the 4 directly below the 3.
• Now rotate the pair: slide the 3 left and bring the 4 up. Both land in one go, and the row is closed.

The exact same shape works for the last tile of every row and, rotated ninety degrees, for the last tile of every column. Learn it once and you can close any grid, at any size.

Advice

• Move the gap, not the tiles. Thinking in terms of "where do I take the hole" makes obvious a lot of manoeuvres that otherwise look like magic.
• Only count moves from your second game. The first one is for learning the method; you improve your record afterwards, by cutting out the gap’s wasted loops.
• The 6×6 changes nothing. It is the same method with more rows: if it works on a 4×4 it works at any size, which is why it is worth learning instead of going on instinct.

Frequently asked questions

Can a grid be impossible?

In the physical puzzle yes, in this version no: since it is scrambled from the solution, you can always walk it back.

How are moves counted?

Every tap that slides something is one move, even if three tiles go at once. So the way to lower your record is to use long pushes: taking the gap from one end of a 4×4 row to the other with a single tap costs one move, with three separate taps it costs three. With the arrow keys, on the other hand, every tile is a move.

Does my record survive if I change phone?

Yes: the best move count for each size is stored on your account, and if you sign in on another device the lower of the two is kept.

Can I leave a grid half done?

Yes. When you leave, the grid stays as it is along with the move count, and next time the start screen offers to continue it. There is only one saved game: starting a new one asks for confirmation first.

More games to try

• Sudoku
• 10×10
• 2048

See the full list of games, or open Pocket Games in your browser: they are free and you don’t need an account.